The next step is to generate sample trajectories using the trajectory resulting from the trajectory
update step. In [1]}, the authors suggested the way
of generating samples around the given trajectory using a time-varying
feedback gain \(K(k)\) . By giving additional samples, we can increase the amount of data
and make the neural network be more robust. Following a similar approach, first, we obtain the feedback gain by a linear time-varying LQR solution with the linearized model
given in (). Then, we choose \(S\)  new initial
states \(x_{\text{init}}^{i,s}\)  around the original ones \(x_{\text{init}}^{i}\) 
by adding Gaussian noise, and generate \(S\)  new trajectories with the
linearized model. This can be formulated as follows:
\(x_{\text{init}}^{i,s} & =x_{\text{init}}^{i}+\epsilon ^{i,s},\epsilon ^{i,s}\sim \mathcal {N}(0,\sigma _{i}^{2}),\nonumber \\u_{k}^{i,s} & =\hat{u}_{k}^{i}+K^{i}(k)(x_{k}^{i,s}-\hat{x}_{k}^{i}),\nonumber \\x_{k+1}^{i,s} & =A_{k}^{i}x_{k}^{i,s}+B_{k}^{i}u_{k}^{i,s}+z_{k}^{i}+\hat{\nu }_{k}^{i}.\) 
